The present invention relates generally to the field of signal sensing, sampling and processing and more particularly to systems and methods designed to sense, sample and process relatively sparse signals having spectral content over a very large frequency range.
In the field of analog and digital communications, a signal is commonly defined as anytime-varying or spatial-varying quantity that conveys information (e.g., an energy signal or power advance).
Sampling and reconstruction processes are commonly utilized in signal sensing, sampling and processing systems to, inter alia, digitize an analog input signal. As defined herein, the term “sampling” relates to the process of converting a continuous input signal into a plurality of discrete signals by observing the value of the input signal at a series of sample points. The term “reconstruction” relates to the formation of the collection of discrete signals back into a continuous signal form.
The rate at which a system samples an input signal is commonly referred to as its “sampling rate” in the art. In most systems, input signals are sampled in accordance with the Nyquist sampling rate in order to ensure accurate reconstruction. Simply stated, the Nyquist sampling theorem states that, if the bandwidth of a received signal is f Hz, then at least two samples per cycle are required to ensure proper reconstruction (i.e., the sampling rate must be at least 2f). If a sub-Nyquist sampling rate is utilized, aliasing may occur during reconstruction. As defined herein, “aliasing” relates to, inter alia, (i) the loss of some frequency information with respect to the original signal when sampled at a slow rate and (ii) the generation of frequency-shifted replicas of a target signal when the digitized signal is reconstructed as a continuous time signal. As will be shown in detail below, signal replicas caused from aliasing often create ambiguities and/or overlap with the target signal, which is highly undesirable.
In view of the aforementioned shortcomings associated with sub-Nyquist sampling, systems traditionally utilize a sampling rate that is in accordance with the Nyquist sampling theorem. However, it has been found that sampling in accordance with the Nyquist sampling theorem introduces a number of notable drawbacks.
As a first drawback, systems using a sampling rate at or above the Nyquist standard are often limited in dynamic range. For instance, if the sampling rate of the sampling device is set too high, the noise produced by the device often rises to a level that significantly compromises the overall performance of the system, which is highly undesirable.
As another drawback, systems using a sampling rate at or above the Nyquist standard often require signal processing algorithms that have a substantial, or even unattainable, level of computational complexity, which is highly undesirable.
As another drawback, systems using a sampling rate at or above the Nyquist standard often impose considerable, often impractical, system requirements with respect to data storage, size, weight, power consumption and cost, which is highly undesirable.
As another drawback, systems using a sampling rate at or above the Nyquist standard often exceed the performance capabilities of known sampling devices in the art. For example, in certain wide bandwidth applications, a relatively high sample rate is required that is presently unattainable by most, if not all, conventional sampling devices, which is highly undesirable.
As another drawback, systems using a sampling rate at or above the Nyquist standard often necessitate the practice of “selective channeling”. As defined herein, selective channeling relates to processing only those signals that are located within a particular channel, or segment, of an entire bandwidth of interest. As can be appreciated, selective channeling can result in removing potentially pertinent signals located within the bandwidth of interest from detection and processing, which is highly undesirable in certain applications (e.g., military applications).
In view of the aforementioned restrictions associated with the Nyquist sampling condition, it is well known in the art for systems to utilize an undersampling approach for sampling signals that are known to contain spectral content over a large frequency range. As defined herein, “undersampling” denotes sampling below the Nyquist rate. Specifically, instead of collecting samples at the Nyquist rate of fNyquist=2fmax samples per index variable unit, samples are collected at the rate of fsample=fNyquist/N samples per index variable unit for some N>1. Accordingly, this approach is commonly referred to in the art as undersampling by a factor of N. Furthermore, when the samples are collected at points uniformly spaced in the index variable, this approach is commonly referred to in the art as uniform undersampling by a factor of N.
As noted above, sub-Nyquist sampling induces aliasing. Accordingly, it is to be understood that systems that rely upon undersampling typically include logic to reduce the deleterious effects associated with aliasing.
For example, compressed sampling is one well-known undersampling approach that resolves aliasing using appropriate logic. Specifically, compressed sensing, also known as compressive sampling, is a technique for acquiring and reconstructing a signal by first randomizing, or scrambling, sensed signals located within a relatively large frequency range that are known to be sparse in nature and then, in turn, reconstructing the randomized signal within its proper location within the frequency spectrum using the known signal randomization information.
Although well-known in the art, the compressed sampling approach is limited for use with very sparse signals that have spectral content over a very large frequency range. As defined herein, “sparse signals” denotes signals with relatively low information density (e.g., relatively few of the spectral components of the signal have significant amplitudes and the amplitudes of remaining spectral content are zero or negligibly small). Signals that fail to comply with the degree of sparseness required by the compressed sampling approach are typically incapable of accurate reconstruction, which is highly undesirable.
It should be noted that additional approaches for resolving ambiguities created from undersampling are known in the art. As an example, in U.S. Pat. No. 7,173,555 to G. M. Raz, there is disclosed a method and system of signal processing that utilizes identified distortion products, such as nonlinear artifacts, to resolve ambiguities associated with sub-Nyquist sampling, the disclosure of which is incorporated herein by reference.